On classical solutions to the Hartree equation

Abstract

This paper is concerned with positive classical solutions to the Hartree equation−Δu=(1|x|n−α⁎up)up−1 in Rn. When n≤2, we show that the equation has no positive solution. When n≥3, we prove that the equation has no positive solution if p<n+αn−2, we also classify all positive solutions to the equation in the critical case p=n+αn−2. The main novelty of this paper is that we cover the full range 0<α<n and −∞<p≤n+αn−2 in our results.